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Mirrors > Home > MPE Home > Th. List > pncan2 | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.) |
Ref | Expression |
---|---|
pncan2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcom 10628 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) | |
2 | 1 | oveq1d 6993 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐵 + 𝐴) − 𝐴) = ((𝐴 + 𝐵) − 𝐴)) |
3 | pncan 10694 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐵 + 𝐴) − 𝐴) = 𝐵) | |
4 | 2, 3 | eqtr3d 2816 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
5 | 4 | ancoms 451 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 (class class class)co 6978 ℂcc 10335 + caddc 10340 − cmin 10672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-po 5327 df-so 5328 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-ltxr 10481 df-sub 10674 |
This theorem is referenced by: subid 10708 pnpcan 10728 pnncan 10730 pncan2d 10802 fzrev3 12792 fzrevral3 12813 fzosubel2 12915 facndiv 13466 bcnp1n 13492 lswccatn0lsw 13757 swrds1 13847 swrdccat3b 13948 swrdccat3bOLD 13949 revccat 13988 trireciplem 15080 psgnunilem2 18388 efgredleme 18631 pjthlem1 23746 uniioombllem3 23892 dyadovol 23900 dvfsumle 24324 qaa 24618 geolim3 24634 pserdv2 24724 logtayl 24947 tanatan 25201 atans2 25213 efrlim 25252 ppidif 25445 ppiub 25485 bposlem9 25573 pntrsumo1 25846 pntpbnd1a 25866 pntpbnd2 25868 pntlemr 25883 axsegconlem10 26418 crctcshwlkn0lem6 27304 pjhthlem1 28952 hst1h 29788 ballotlem2 31392 ballotlemfmpn 31398 lzenom 38762 acongrep 38973 fouriersw 41948 |
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